Newton type methods for solving a Hasegawa–Mima plasma model

In Karakazian and Nassif (2019), the non-linear space-time Hasegawa–Mima plasma equation is formulated as a coupled system of two linear PDEs, a solution of which is a pair (u,w), with w=(I−Δ)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa–Mima problem with periodic boundary conditions were also derived. In a more recent work Karakazian et al. (2022), a numerical approach consisting of a finite element space-domain combined with an Euler-implicit time scheme was used to discretize the coupled variational Hasegawa–Mima model. A semi-linear version of this implicit nonlinear scheme was tested for several types of initial conditions. This semi-linear scheme proved to lack efficiency for long time, which necessitates imposing a cap on the magnitude of the solution. To circumvent this difficulty, in this paper, we use Newton-type methods (Newton, Chord and an introduced Modified Newton method) to solve numerically the fully-implicit non-linear scheme. Testing these methods in FreeFEM++ indicates significant improvements as no cap needs to be imposed for long time. In the sequel, we demonstrate the validity of these methods by proving several results, in particular the convergence of the implemented methods.